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Reasoning is a fundamental aspect of mathematics. Reasoning acts basic role and capability to improve general mathematical abilities. In mathematics, reasoning is utilized to solve problems and also to decide whether an assertion (e.g. an answer to a problem) is correct. The power of mathematics lies in relation and transformation which give rise to pattern and generalization. Generalization implies such deliberate reasoning that builds on specific cases to identify inter-model, inter-procedural or inter-structural relationships. Generalization plays a crucial role in the activity of any mathematician, being considered an inherent ability to mathematical thinking in general. Patterns is a fundamental stage in the formation of generalization. Pattern generalization is a core area in mathematics that is recognized by a community of researchers as an approach to develop student’s algebraic reasoning. Patterning is critical to the abstraction of mathematical ideas and relationships, and the development of mathematical reasoning in young children. Examining students’ generalization strategies of patterns becomes very important in term of learning advanced algebra.

 

This study applied qualitative descriptive methods and was to investigate of students’ reasoning behavior to generalize some algebraic patterns. An instrument was designed for this study and was validated by two mathematics lecturers. Four problems related to figural patterns were included as a tool of data collection. Each problem consisted of six questions. The questions were based on the type of generalization. The type of generalization was immediate generalization, near generalization, and far generalization. Immediate generalization was addressed in point a, b in each problems. Near generalization was referred to point c, d in each problems. Far generalization was related to point e, f in each problems. The sample were the 8^th grade students of SMPN 2 Jatirejo Mojokerto. All students were given the algebraic problems related to figural patterns from which the results were categorized into three mathematics levels, i.e. high, medium, and low mathematics level respectively. Three volunteer students (first, second, and third) were selected as research subjetcs and based on the existing of students reasoning behaviors. The semi-structured interview were individually conducted to reveal students reasoning behaviors in constructing generalize patterns while they solved the given problem. The obtained data was classified through the generalization strategies in the existing literatures.

 

According to the results of the task performed by the first subject, she used recursive strategies for immediate generalization on each problem. She was able to recognize figural pattern by finding the difference between the given pattern. Then she added the difference into the given pattern to determine the next pattern. She also used recursive strategies for near generalization on each problem but she had such difficulty in explaining the answers to problems 3 and 4. She utilized functional strategies for far generalization on each problem but had an incorrectly answer. She identified the growing components of the pattern as well as the constant components but she was not insert the constant component correctly on the rule. Overall, she performed figural reasoning based on the figural pattern task.

 

Similarly, the second subject implemented recursive strategies for immediate generalization on each problem. Take into account that the first subject recognized figural pattern by finding the difference between the given pattern,  the second subject involved numerical pattern. She summed the difference up to the given number from figural pattern to obtain the next pattern. She also employed recursive strategies for near generalization on each problem. She implied functional strategies for far generalization on each problem but had an incorrect answer. She was able to identify the growing components of the pattern as well as the constant components but she was not insert the constant component correctly on the rule. Overall, she categorized numerical reasoning based on the figural pattern task.

 

Moreover, the third subject did not use strategy on the framework. On problem 1, she recognized the pattern as an odd number starting from number 3. Then she listed the odd number from figure 1 to figure 19 in order to answer the question on points a through d. On problem 2, she had performed the pattern as a multiple of 4 starting from number 6. Then she registered the odd number from figure 1 to figure 19 to answer the question on points a through d. On problems 3 and 4, she also considered the figural pattern as a multiple of odd numbers. Overall, she had clearly answered all questions but involved no strategy on the given framework.

 

The result of this study emphasized that students mainly utilized a recursive strategy to achieve immediate and near generalization. All subjects answered the problems correctly. Two of subjects used functional strategy to reach the far generalization but had not yet received the correct answer. The expected implications of this study are as consideration for designing the learning on the generalization of patterns in accordance with the students' reasoning behavior

reasoning behavior, generalization, figural pattern, immediate generalization, near generalization, far generalization